Module 3

Tropical Cyclones & Hurricanes

Thermodynamic engines of destruction: from sea-surface heat to Category 5 wind

3.1 Emanuel Potential Intensity (PI) Theory

Kerry Emanuel's breakthrough insight (1986, 1988) was to model a mature tropical cyclone as a Carnot heat engine. The warm ocean surface is the hot reservoir, and the cold tropopause (\(\sim\)200 hPa outflow level) is the cold reservoir. The working fluid is moist air that picks up enthalpy at the surface and exports it aloft via the outflow layer.

The Carnot Cycle Analogy

The thermodynamic cycle has four legs:

Isothermal expansion at the sea surface: air spirals inward at constant SST, gaining enthalpy from ocean evaporation (\(T_{s}\))
Adiabatic ascent in the eyewall: moist air rises pseudoadiabatically, converting enthalpy to kinetic and potential energy
Isothermal compression at the outflow: air radiates to space near tropopause temperature (\(T_{o}\))
Adiabatic descent in the environment: air subsides back toward the surface

Deriving the PI Equation

The Carnot efficiency for a heat engine operating between \(T_s\) and \(T_o\) is:

\( \eta = \frac{T_s - T_o}{T_o} \)

Note the denominator is \(T_o\) (not \(T_s\)) because dissipative heating at the surface feeds back into the system. The energy input is the enthalpy flux from the ocean surface. The enthalpy difference between the saturated boundary layer air at SST and the ambient boundary layer air is:

\( \Delta h = h_0^* - h \)

where \(h_0^*\) is the saturation moist static energy at SST and \(h\)is the actual moist static energy of the boundary layer air. The surface enthalpy flux follows a bulk aerodynamic formula with exchange coefficient \(C_k\), while the surface stress (momentum sink) depends on drag coefficient \(C_d\). Balancing energy input against frictional dissipation:

\( V_{\max}^2 = \frac{C_k}{C_d} \cdot \frac{T_s - T_o}{T_o} \cdot (h_0^* - h) \)

This is the Emanuel Potential Intensity equation. Each factor has a clear physical interpretation:

\(C_k/C_d\): ratio of enthalpy exchange to momentum exchange (\(\approx 0.9\text{--}1.1\) over tropical oceans)
\((T_s - T_o)/T_o\): thermodynamic efficiency, typically \(\sim 1/3\) for \(T_s \approx 300\) K, \(T_o \approx 200\) K
\(h_0^* - h\): thermodynamic disequilibrium between ocean and atmosphere (\(\sim 10\text{--}20\) kJ/kg)

Why PI Increases with SST

The saturation specific humidity \(q^*\) increases exponentially with temperature via the Clausius-Clapeyron relation (\(\sim 7\%/°\text{C}\)), so \(h_0^*\)grows rapidly with SST. Since \(T_o\) changes little (the tropopause temperature is relatively stable), both the efficiency and the enthalpy disequilibrium increase with warming. The net effect is that PI increases by approximately \(3\text{--}5\%\) per degree of SST warming, or equivalently a \(\sim 6\text{--}10\%\) increase in destructive potential (\(\propto V^3\)).

3.2 Eyewall Dynamics & Structure

The eyewall is the ring of deepest convection surrounding the calm eye. Understanding its structure requires combining gradient wind balance, angular momentum conservation, and Ekman pumping.

Gradient Wind Balance

In the free atmosphere above the boundary layer, the radial equation of motion for an axisymmetric vortex in cylindrical coordinates \((r, \phi, z)\):

\( \frac{v^2}{r} + fv = \frac{1}{\rho}\frac{\partial p}{\partial r} \)

where \(v\) is the tangential wind, \(f\) is the Coriolis parameter, and the right-hand side is the radial pressure gradient force. The centrifugal (\(v^2/r\)) and Coriolis (\(fv\)) accelerations balance the inward-directed pressure gradient.

Angular Momentum & Radius of Maximum Wind

The absolute angular momentum per unit mass is:

\( M = rv + \frac{1}{2}fr^2 \)

In the free atmosphere above the boundary layer, \(M\) is approximately conserved following parcels. Air that spirals inward from large radius \(r_0\) (where\(v \approx 0\)) carries angular momentum \(M_0 = \frac{1}{2}fr_0^2\). At the radius of maximum wind \(r_{\max}\):

\( V_{\max} = \frac{M_0}{r_{\max}} - \frac{1}{2}fr_{\max} = \frac{fr_0^2}{2r_{\max}} - \frac{fr_{\max}}{2} \)

This shows that stronger storms (larger \(V_{\max}\)) require either a larger source radius \(r_0\) or a smaller \(r_{\max}\). In practice, the radius of maximum wind for intense hurricanes is typically 15\(\text{--}\)40 km.

Ekman Pumping & Secondary Circulation

In the boundary layer, surface friction breaks the gradient wind balance, causing air to spiral inward. The convergence at the top of the boundary layer drives upward motion (Ekman pumping):

\( w_E = \frac{1}{\rho} \nabla \times \left(\frac{\boldsymbol{\tau}_s}{\rho f}\right) \approx \frac{\zeta}{2} \sqrt{\frac{\nu}{f}} \)

where \(\zeta\) is the relative vorticity and \(\nu\) is the eddy viscosity. The secondary (overturning) circulation consists of: (1) frictional inflow in the boundary layer, (2) forced ascent in the eyewall convection, and (3) outflow near the tropopause. This secondary circulation imports angular momentum into the vortex, intensifying the primary (tangential) circulation.

3.3 Rapid Intensification

Rapid intensification (RI) is defined as a sustained wind speed increase of \(\geq 30\) knots (15.4 m/s) in 24 hours. Events like Hurricane Patricia (2015), which intensified by 120 knots in 24 hours, demonstrate that the atmosphere can approach PI on extremely short timescales. RI events have become more frequent under climate change, posing enormous forecasting challenges.

Conditions Favouring RI

High SST (\(\geq 26.5°\text{C}\), ideally \(\geq 28.5°\text{C}\)): provides the enthalpy reservoir
Deep warm water: ocean heat content (OHC) to 26°C isotherm depth > 50 kJ/cm\(^2\) prevents upwelling-induced cooling
Low vertical wind shear: \(|\Delta \mathbf{V}_{200-850}| < 10\) m/s to keep the vortex vertically aligned
Moist mid-levels: relative humidity at 500\(\text{--}\)700 hPa \(\geq 70\%\) minimises entrainment dilution of convective updrafts
Upper-level outflow: anticyclonic flow aloft evacuates mass efficiently

The Ventilation Index

Tang and Emanuel (2012) defined a ventilation indexthat captures the competition between environmental shear (which imports low-entropy air into the core) and the storm's thermodynamic potential:

\( \Lambda = \frac{V_{\text{shear}} \cdot \chi_m}{\text{PI}} \)

where \(V_{\text{shear}}\) is the 200\(\text{--}\)850 hPa shear magnitude, \(\chi_m\)is the non-dimensional mid-level entropy deficit:

\( \chi_m = \frac{s_m^* - s_m}{s_{\text{sfc}}^* - s_{\text{sfc}}} \)

where \(s_m^*\) is the saturated entropy at mid-levels, \(s_m\) is the actual mid-level entropy, and the denominator is the surface entropy disequilibrium. Low \(\Lambda\) (\(\lesssim 0.1\)) favours intensification; high\(\Lambda\) (\(\gtrsim 0.3\)) indicates the storm is being ventilated (weakened) by environmental shear. RI typically occurs when \(\Lambda < 0.1\)and the storm is well below its PI.

3.4 Saffir-Simpson Scale & Accumulated Cyclone Energy

Saffir-Simpson Hurricane Wind Scale

Accumulated Cyclone Energy (ACE)

ACE integrates the kinetic energy of all named storms over a season. For a single storm tracked at 6-hourly intervals:

\( \text{ACE} = 10^{-4} \sum_i v_{\max,i}^2 \)

where \(v_{\max,i}\) is the maximum sustained wind (knots) at each 6-hourly observation and the sum runs over all periods when \(v_{\max} \geq 35\) knots. The \(10^{-4}\) factor keeps values manageable. A typical Atlantic season has ACE \(\sim 100\); hyperactive seasons exceed 200.

Under warming, ACE increases because: (1) higher PI allows stronger storms (\(v_{\max}^2\) grows), (2) storms maintain intensity longer over warmer oceans, and (3) the tropical cyclone season may lengthen. As explored in ourClimate & Biodiversity course, these changes have cascading effects on marine and terrestrial ecosystems.

3.5 Storm Surge Physics

Storm surge is the abnormal rise of water generated by a tropical cyclone. It is the leading cause of hurricane fatalities. Two dominant mechanisms contribute:

Inverse Barometer Effect

Reduced atmospheric pressure at the storm centre allows the ocean surface to rise hydrostatically:

\( \eta_{\text{IB}} = \frac{\Delta P}{\rho_w g} \)

where \(\Delta P = P_{\infty} - P_{\text{centre}}\) is the pressure deficit,\(\rho_w \approx 1025\) kg/m\(^3\), and \(g = 9.81\) m/s\(^2\). For a Category 5 hurricane with \(\Delta P \approx 100\) hPa:\(\eta_{\text{IB}} \approx 10000/(1025 \times 9.81) \approx 1.0\) m.

Wind Setup

Onshore winds exert a surface stress \(\tau_s\) that pushes water toward the coast. For a steady-state balance over a shallow shelf of length \(L\) and depth \(D\):

\( \eta_{\text{wind}} = \frac{\tau_s \cdot L}{\rho_w g D} \)

The wind stress is \(\tau_s = \rho_a C_d V_{10}^2\), where \(C_d \approx 2.5 \times 10^{-3}\)at hurricane-force winds. The total surge height:

\( \eta = \eta_{\text{IB}} + \eta_{\text{wind}} = \frac{\Delta P}{\rho_w g} + \frac{\tau_s L}{\rho_w g D} \)

Wind setup typically dominates, producing surges of 3\(\text{--}\)9 m for major hurricanes making landfall over shallow continental shelves. Storm surge is exacerbated by sea-level rise, creating compound flooding risk when combined with extreme precipitation (seeModule 4).

3.6 Cyclone, Hurricane, Typhoon: Same Physics, Different Basins

The same phenomenon receives different names by basin: hurricane (North Atlantic, eastern North Pacific), typhoon (western North Pacific), cyclone (Indian Ocean, South Pacific). All require \(f \neq 0\) to generate initial rotation via the Coriolis effect.

The Coriolis parameter \(f = 2\Omega\sin\phi\) vanishes at the equator (\(\phi = 0\)), which is why tropical cyclones never form within\(\sim 5°\) of the equator. The minimum latitude provides sufficient\(f\) for the initial vortex to organise. In the Atmospheric Dynamics module we derived the Coriolis force in detail.

The rotation direction is set by Coriolis: counterclockwise in the Northern Hemisphere, clockwise in the Southern Hemisphere. This is a direct consequence of angular momentum conservation on a rotating sphere.

The Science of Hurricanes — NASA Goddard

Key Concepts in This Module

•Emanuel Potential Intensity theory (TC as Carnot engine)

•Gradient wind balance and angular momentum in eyewall

•Rapid intensification criteria and ventilation index

•Storm surge physics: inverse barometer + wind setup

•ACE trends and climate change projections

•Coriolis constraint on TC formation latitude

3.7 Warm Core Structure & Thermal Wind

Tropical cyclones are warm-core systems: the temperature anomaly in the eye and eyewall region is positive relative to the environment. This warmth comes from latent heat release in the eyewall convection and adiabatic warming from subsidence in the eye. The thermal wind relation connects the warm core to the wind structure:

\( \frac{\partial v}{\partial \ln p} = -\frac{R}{f + 2v/r}\frac{\partial T}{\partial r} \)

A warm core (\(\partial T/\partial r < 0\) inward) requires that the tangential wind \(v\) decreases with height above the boundary layer. This is exactly what is observed: the maximum winds are near the surface, and the vortex weakens aloft. The magnitude of the warm core anomaly for a Category 5 hurricane can exceed\(+15°\text{C}\) at 300\(\text{--}\)200 hPa.

The warm core also produces the characteristic central pressure minimum. Integrating the hypsometric equation from surface to the tropopause with a warmer-than-environment column:

\( \Delta p_{\text{sfc}} \approx \frac{p_{\text{sfc}}}{H} \int_0^H \frac{\Delta T(z)}{T_{\text{env}}(z)} dz \)

For a strong hurricane with a deep warm core anomaly of \(+10°\text{C}\) through a 15 km deep troposphere, this produces a surface pressure deficit of approximately\(80\text{--}100\) hPa, consistent with observations of intense storms.

Tropical Cyclone Cross-Section

Schematic vertical cross-section showing the eye, eyewall, spiral rainbands, and the thermodynamic circulation (inflow, ascent, outflow):

Simulation: Potential Intensity vs SST

Computing the theoretical maximum wind speed as a function of sea surface temperature using the simplified Emanuel PI equation. Notice the rapid increase above 26°C:

Emanuel Potential Intensity vs SST

Python

Theoretical maximum TC wind speed as a function of sea surface temperature

script.py87 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Emanuel PI: V_max^2 = (Ck/Cd) * (Ts - To)/To * (h0* - h)
# h0* - h is dominated by Lv * (q*(Ts) - q)
# Use Clausius-Clapeyron for q*(T)

def sat_specific_humidity(T_C, P=1013.25):
    """Saturation specific humidity (kg/kg) from T in Celsius"""
    T_K = T_C + 273.15
    # Bolton (1980) formula for saturation vapor pressure (hPa)
    e_s = 6.112 * np.exp(17.67 * T_C / (T_C + 243.5))
    q_s = 0.622 * e_s / (P - 0.378 * e_s)
    return q_s

# Parameters
Ck_Cd = 0.95       # ratio of exchange coefficients
To = 200.0          # outflow temperature (K)
Lv = 2.5e6          # latent heat of vaporization (J/kg)
cp = 1005.0         # specific heat at constant pressure
RH_bl = 0.80        # boundary layer relative humidity

# SST range
SST = np.linspace(20, 35, 200)
Ts = SST + 273.15   # in Kelvin

# Enthalpy disequilibrium: Lv * (q*(Ts) - q_bl)
q_star = sat_specific_humidity(SST)
q_bl = RH_bl * sat_specific_humidity(SST - 2)  # BL air is cooler
delta_h = Lv * (q_star - q_bl) + cp * 2  # enthalpy diff (sensible + latent)

# Carnot efficiency
eta = (Ts - To) / To

# PI
Vmax_sq = Ck_Cd * eta * delta_h
Vmax_sq = np.maximum(Vmax_sq, 0)
Vmax = np.sqrt(Vmax_sq)

# Convert to knots
Vmax_kt = Vmax * 1.944

# Saffir-Simpson thresholds (knots)
cat_thresholds = [64, 83, 96, 113, 137]
cat_labels = ['Cat 1', 'Cat 2', 'Cat 3', 'Cat 4', 'Cat 5']
cat_colors = ['#44aa88', '#88cc44', '#cccc44', '#ff8844', '#ff4444']

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a1a')

# Left panel: PI vs SST
ax1.set_facecolor('#0a0a2e')
ax1.plot(SST, Vmax_kt, color='#6688ff', linewidth=2.5, label='Potential Intensity')
for i, (thresh, label, color) in enumerate(zip(cat_thresholds, cat_labels, cat_colors)):
    ax1.axhline(y=thresh, color=color, linewidth=1, linestyle='--', alpha=0.7)
    ax1.text(20.5, thresh + 1.5, label, color=color, fontsize=9, fontweight='bold')

# Mark the 26.5C threshold
ax1.axvline(x=26.5, color='#ffaa44', linewidth=1.5, linestyle=':', alpha=0.8)
ax1.text(26.7, 30, 'SST threshold\n26.5°C', color='#ffaa44', fontsize=9)

ax1.set_xlabel('Sea Surface Temperature (°C)', color='white', fontsize=12)
ax1.set_ylabel('Maximum Wind Speed (knots)', color='white', fontsize=12)
ax1.set_title('Emanuel Potential Intensity vs SST', color='white', fontsize=14, fontweight='bold')
ax1.tick_params(colors='white')
ax1.set_xlim(20, 35)
ax1.set_ylim(0, 200)
ax1.grid(True, alpha=0.2, color='#4444aa')
ax1.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white')

# Right panel: V_max vs Carnot efficiency
ax2.set_facecolor('#0a0a2e')
ax2.plot(eta, Vmax, color='#aa66ff', linewidth=2.5)
ax2.set_xlabel('Carnot Efficiency (T_s - T_o)/T_o', color='white', fontsize=12)
ax2.set_ylabel('Maximum Wind Speed (m/s)', color='white', fontsize=12)
ax2.set_title('PI Dependence on Thermodynamic Efficiency', color='white', fontsize=14, fontweight='bold')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.2, color='#4444aa')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
print("PI ranges from {:.0f} to {:.0f} knots over SST {:.0f}-{:.0f}°C".format(Vmax_kt[0], Vmax_kt[-1], SST[0], SST[-1]))
print("At SST=28°C: V_max = {:.0f} knots".format(np.interp(28, SST, Vmax_kt)))
print("At SST=30°C: V_max = {:.0f} knots".format(np.interp(30, SST, Vmax_kt)))
print("Fractional increase per °C near 28°C: {:.1f}%".format(100*(np.interp(29,SST,Vmax_kt)-np.interp(28,SST,Vmax_kt))/np.interp(28,SST,Vmax_kt)))

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation: Holland Radial Wind Profile

The Holland (1980) parametric model gives the radial wind profile of a tropical cyclone. The pressure profile is:

\( P(r) = P_c + \Delta P \exp\!\left[-\left(\frac{R_{\max}}{r}\right)^B\right] \)

where \(B\) is the Holland shape parameter controlling the peakedness of the wind profile. The gradient wind then follows from the radial pressure gradient:

\( V(r) = \sqrt{\frac{B}{\rho}\left(\frac{R_{\max}}{r}\right)^B \Delta P \exp\!\left[-\left(\frac{R_{\max}}{r}\right)^B\right] + \frac{r^2 f^2}{4}} - \frac{rf}{2} \)

Holland Parametric Wind Profile

Python

Radial wind structure for different storm categories and Holland B parameters

script.py63 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

def holland_wind(r, Rmax, dP, B, rho=1.15, f=5e-5):
    """Holland (1980) gradient wind profile"""
    x = (Rmax / r) ** B
    term1 = (B / rho) * x * dP * np.exp(-x)
    term2 = (r * f / 2) ** 2
    V = np.sqrt(term1 + term2) - r * f / 2
    return np.maximum(V, 0)

r = np.linspace(1, 500, 1000)  # radius in km
r_m = r * 1000  # convert to meters

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a1a')

# Left: Different storm intensities
ax1.set_facecolor('#0a0a2e')
storms = [
    {'name': 'Cat 1 (dP=30 hPa)', 'dP': 3000, 'Rmax': 50e3, 'B': 1.5, 'color': '#44aa88'},
    {'name': 'Cat 3 (dP=60 hPa)', 'dP': 6000, 'Rmax': 35e3, 'B': 1.7, 'color': '#cccc44'},
    {'name': 'Cat 5 (dP=100 hPa)', 'dP': 10000, 'Rmax': 20e3, 'B': 2.0, 'color': '#ff4444'},
]

for s in storms:
    V = holland_wind(r_m, s['Rmax'], s['dP'], s['B'])
    V_kt = V * 1.944
    ax1.plot(r, V_kt, color=s['color'], linewidth=2.5, label=s['name'])
    # Mark Vmax
    idx = np.argmax(V_kt)
    ax1.plot(r[idx], V_kt[idx], 'o', color=s['color'], markersize=8)

ax1.set_xlabel('Radius (km)', color='white', fontsize=12)
ax1.set_ylabel('Wind Speed (knots)', color='white', fontsize=12)
ax1.set_title('Wind Profile by Storm Intensity', color='white', fontsize=14, fontweight='bold')
ax1.tick_params(colors='white')
ax1.set_xlim(0, 300)
ax1.grid(True, alpha=0.2, color='#4444aa')
ax1.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=10)

# Right: Effect of Holland B parameter
ax2.set_facecolor('#0a0a2e')
for B_val, color, ls in [(1.0, '#4488ff', '-'), (1.5, '#88aaff', '--'), (2.0, '#aaccff', '-'), (2.5, '#ccddff', '--')]:
    V = holland_wind(r_m, 30e3, 7000, B_val)
    V_kt = V * 1.944
    ax2.plot(r, V_kt, color=color, linewidth=2, linestyle=ls, label=f'B = {B_val}')

ax2.set_xlabel('Radius (km)', color='white', fontsize=12)
ax2.set_ylabel('Wind Speed (knots)', color='white', fontsize=12)
ax2.set_title('Effect of Holland B Parameter', color='white', fontsize=14, fontweight='bold')
ax2.tick_params(colors='white')
ax2.set_xlim(0, 300)
ax2.grid(True, alpha=0.2, color='#4444aa')
ax2.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=10)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
print("Holland wind profile computed successfully.")
print("Higher B = sharper peak, more compact storm.")
print("Cat 5 peak at R_max = 20 km; Cat 1 peak at R_max = 50 km.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation: Storm Surge Model

Computing surge height from inverse barometer effect and wind setup for varying storm intensity and shelf geometry:

Storm Surge: Inverse Barometer + Wind Setup

Python

Surge height components as a function of storm intensity and shelf properties

script.py101 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Physical constants
rho_w = 1025.0   # seawater density (kg/m^3)
rho_a = 1.15     # air density (kg/m^3)
g = 9.81         # gravity (m/s^2)
Cd = 2.5e-3      # drag coefficient at hurricane winds

def storm_surge(dP_hPa, V10, L_shelf, D_shelf):
    """Compute surge height (m)
    dP_hPa: pressure deficit in hPa
    V10: 10-m wind speed (m/s)
    L_shelf: shelf length (m)
    D_shelf: shelf depth (m)
    """
    dP = dP_hPa * 100  # convert to Pa
    eta_IB = dP / (rho_w * g)
    tau_s = rho_a * Cd * V10**2
    eta_wind = tau_s * L_shelf / (rho_w * g * D_shelf)
    return eta_IB, eta_wind, eta_IB + eta_wind

fig, axes = plt.subplots(1, 3, figsize=(16, 5.5))
fig.patch.set_facecolor('#0a0a1a')

# Panel 1: Surge vs wind speed
ax = axes[0]
ax.set_facecolor('#0a0a2e')
V = np.linspace(20, 80, 200)  # m/s
dP_vals = 2.5 * V  # rough dP-V relationship (hPa)

for L, D, label, color in [(200e3, 30, 'Wide shallow (200km, 30m)', '#ff6644'),
                             (100e3, 50, 'Medium (100km, 50m)', '#ffaa44'),
                             (50e3, 100, 'Narrow deep (50km, 100m)', '#44aaff')]:
    surges = [storm_surge(dp, v, L, D)[2] for dp, v in zip(dP_vals, V)]
    ax.plot(V * 1.944, surges, color=color, linewidth=2.5, label=label)

# Category markers
for cat, v_kt, color in [('C1', 64, '#44aa88'), ('C3', 96, '#cccc44'), ('C5', 137, '#ff4444')]:
    ax.axvline(x=v_kt, color=color, linewidth=1, linestyle=':', alpha=0.6)
    ax.text(v_kt+1, 0.3, cat, color=color, fontsize=9)

ax.set_xlabel('Wind Speed (knots)', color='white', fontsize=11)
ax.set_ylabel('Surge Height (m)', color='white', fontsize=11)
ax.set_title('Surge vs Wind Speed', color='white', fontsize=13, fontweight='bold')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.2, color='#4444aa')
ax.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=8)

# Panel 2: Components for a Cat 4 storm
ax = axes[1]
ax.set_facecolor('#0a0a2e')
L_range = np.linspace(20e3, 300e3, 200)
eta_IB, eta_wind, eta_total = [], [], []
for L in L_range:
    ib, w, t = storm_surge(80, 60, L, 40)  # Cat 4 conditions
    eta_IB.append(ib)
    eta_wind.append(w)
    eta_total.append(t)

ax.fill_between(L_range/1000, 0, eta_IB, alpha=0.4, color='#6688ff', label='Inverse Barometer')
ax.fill_between(L_range/1000, eta_IB, eta_total, alpha=0.4, color='#ff8844', label='Wind Setup')
ax.plot(L_range/1000, eta_total, color='#ffffff', linewidth=2, label='Total')

ax.set_xlabel('Continental Shelf Width (km)', color='white', fontsize=11)
ax.set_ylabel('Surge Height (m)', color='white', fontsize=11)
ax.set_title('Surge Components (Cat 4)', color='white', fontsize=13, fontweight='bold')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.2, color='#4444aa')
ax.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=9)

# Panel 3: Effect of sea level rise on surge
ax = axes[2]
ax.set_facecolor('#0a0a2e')
SLR = np.array([0, 0.3, 0.5, 0.8, 1.0, 1.5])  # m
cats = [('Cat 1', 30, 33, '#44aa88'), ('Cat 3', 60, 50, '#cccc44'), ('Cat 5', 100, 70, '#ff4444')]

for name, dP, V10, color in cats:
    _, _, base_surge = storm_surge(dP, V10, 150e3, 40)
    total_with_slr = base_surge + SLR
    ax.plot(SLR, total_with_slr, 'o-', color=color, linewidth=2, markersize=5, label=name)

ax.set_xlabel('Sea Level Rise (m)', color='white', fontsize=11)
ax.set_ylabel('Total Water Level (m)', color='white', fontsize=11)
ax.set_title('Surge + Sea Level Rise', color='white', fontsize=13, fontweight='bold')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.2, color='#4444aa')
ax.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=10)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')

# Print summary
ib, w, t = storm_surge(80, 60, 150e3, 40)
print(f"Cat 4 storm surge (150km shelf, 40m depth):")
print(f"  Inverse barometer: {ib:.2f} m")
print(f"  Wind setup: {w:.2f} m")
print(f"  Total: {t:.2f} m")
print(f"  With 1m SLR: {t+1:.2f} m")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation: ACE Trends 1950\(\text{--}\)Present

Modelling the expected trend in Accumulated Cyclone Energy given observed SST warming and the PI-SST relationship. We use synthetic data calibrated to observations:

Accumulated Cyclone Energy Trends

Python

Modelled ACE trends for North Atlantic and Western Pacific basins

script.py108 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

np.random.seed(42)

years = np.arange(1950, 2026)
n = len(years)

# Simulate SST trend (observed ~0.13°C/decade in tropical Atlantic)
sst_trend = 0.013 * (years - 1950)
sst_variability = 0.3 * np.sin(2 * np.pi * (years - 1950) / 60)  # AMO-like
sst_noise = np.random.normal(0, 0.15, n)
sst_anom = sst_trend + sst_variability + sst_noise

# ACE model: ACE ~ exp(alpha * SST_anom) * base_ACE
# With ENSO modulation and random variability
alpha = 0.4  # ACE sensitivity to SST
base_ace_atl = 100  # baseline Atlantic ACE

# ENSO index (synthetic)
enso = 1.5 * np.sin(2 * np.pi * (years - 1950) / 4.5 + np.random.uniform(0, 2*np.pi))
enso += np.random.normal(0, 0.5, n)

# Atlantic ACE (La Nina tends to increase it)
ace_atl = base_ace_atl * np.exp(alpha * sst_anom - 0.15 * enso)
ace_atl += np.random.normal(0, 20, n)
ace_atl = np.maximum(ace_atl, 10)

# Western Pacific ACE (El Nino tends to increase it)
base_ace_wpac = 300
ace_wpac = base_ace_wpac * np.exp(0.3 * sst_anom + 0.1 * enso)
ace_wpac += np.random.normal(0, 40, n)
ace_wpac = np.maximum(ace_wpac, 50)

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0a0a1a')

# Top left: Atlantic ACE
ax = axes[0, 0]
ax.set_facecolor('#0a0a2e')
ax.bar(years, ace_atl, color='#4466cc', alpha=0.7, width=0.8)
# 10-year running mean
kernel = np.ones(10) / 10
ace_smooth = np.convolve(ace_atl, kernel, mode='same')
ace_smooth[:5] = np.nan
ace_smooth[-5:] = np.nan
ax.plot(years, ace_smooth, color='#ff6644', linewidth=2.5, label='10-yr mean')
ax.axhline(y=100, color='#44aa88', linewidth=1, linestyle='--', alpha=0.5, label='1981-2010 median')
ax.set_ylabel('ACE (10⁴ kt²)', color='white', fontsize=11)
ax.set_title('North Atlantic ACE', color='white', fontsize=13, fontweight='bold')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15, color='#4444aa')
ax.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=9)

# Top right: W Pacific ACE
ax = axes[0, 1]
ax.set_facecolor('#0a0a2e')
ax.bar(years, ace_wpac, color='#cc4466', alpha=0.7, width=0.8)
ace_smooth_wp = np.convolve(ace_wpac, kernel, mode='same')
ace_smooth_wp[:5] = np.nan
ace_smooth_wp[-5:] = np.nan
ax.plot(years, ace_smooth_wp, color='#ffaa44', linewidth=2.5, label='10-yr mean')
ax.set_ylabel('ACE (10⁴ kt²)', color='white', fontsize=11)
ax.set_title('Western Pacific ACE', color='white', fontsize=13, fontweight='bold')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15, color='#4444aa')
ax.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=9)

# Bottom left: SST anomaly
ax = axes[1, 0]
ax.set_facecolor('#0a0a2e')
ax.fill_between(years, 0, sst_anom, where=sst_anom >= 0, color='#ff4444', alpha=0.5)
ax.fill_between(years, 0, sst_anom, where=sst_anom < 0, color='#4444ff', alpha=0.5)
ax.plot(years, sst_trend, color='#ffffff', linewidth=2, linestyle='--', label='Linear trend')
ax.set_xlabel('Year', color='white', fontsize=11)
ax.set_ylabel('SST Anomaly (°C)', color='white', fontsize=11)
ax.set_title('Tropical Atlantic SST Anomaly', color='white', fontsize=13, fontweight='bold')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15, color='#4444aa')
ax.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=9)

# Bottom right: Cat 4-5 fraction
frac_intense = 0.25 + 0.15 * np.tanh(0.03 * (years - 1990))
frac_intense += np.random.normal(0, 0.05, n)
frac_intense = np.clip(frac_intense, 0.05, 0.65)

ax = axes[1, 1]
ax.set_facecolor('#0a0a2e')
ax.scatter(years, frac_intense * 100, color='#aa66ff', s=15, alpha=0.7)
frac_smooth = np.convolve(frac_intense, kernel, mode='same') * 100
frac_smooth[:5] = np.nan
frac_smooth[-5:] = np.nan
ax.plot(years, frac_smooth, color='#ff66aa', linewidth=2.5, label='10-yr mean')
ax.set_xlabel('Year', color='white', fontsize=11)
ax.set_ylabel('Fraction (%)', color='white', fontsize=11)
ax.set_title('Cat 4-5 Fraction of All TCs', color='white', fontsize=13, fontweight='bold')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15, color='#4444aa')
ax.legend(facecolor='#1a1a3e', edgecolor='#4444aa', labelcolor='white', fontsize=9)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')

print(f"Atlantic ACE trend: {np.polyfit(years, ace_atl, 1)[0]:.1f} per year")
print(f"Cat 4-5 fraction increased from ~{frac_intense[0]*100:.0f}% to ~{frac_intense[-1]*100:.0f}%")
print(f"SST warming: {sst_trend[-1]:.2f}°C over {n} years")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

3.7 Eyewall Replacement Cycles & Concentric Eyewalls

Intense tropical cyclones (typically Category 4\(\text{--}\)5) often undergoeyewall replacement cycles (ERCs). An outer rainband organises into a secondary (concentric) eyewall that contracts inward, choking off the moisture supply to the inner eyewall. The process follows a characteristic sequence:

Formation: outer convective ring organises at \(r \sim 2\text{--}3 \times R_{\max}\), often triggered by external forcing (upper-level trough, beta-shear)
Intensification of outer eyewall: the outer ring contracts and intensifies, developing its own tangential wind maximum
Weakening phase: the inner eyewall is cut off from boundary-layer inflow and weakens. The storm temporarily weakens by 10\(\text{--}\)30 knots
Replacement: the outer eyewall contracts to become the new primary eyewall. The storm may re-intensify, often with a larger \(R_{\max}\)

ERCs are a major source of intensity forecast error. The temporary weakening can provide a false sense of safety before re-intensification. Satellite microwave imagery has made ERC detection more reliable, but predicting their timing remains challenging.

The dynamics can be understood through vortex Rossby wavetheory. The radial vorticity gradient supports wave propagation that can organise convection into spiral bands. When the vorticity gradient is sufficiently sharp (associated with the outer wind profile), axisymmetrisation of asymmetric perturbations leads to secondary eyewall formation. The relevant parameter is the filamentation time:

\( \tau_{\text{fil}} = \frac{1}{\sqrt{S_1^2 + S_2^2 - \zeta^2/4}} \)

where \(S_1\) and \(S_2\) are the strain rate components and\(\zeta\) is the relative vorticity. In the rapid filamentation zone (short\(\tau_{\text{fil}}\)), convective cells are stretched and suppressed, creating the moat region between inner and outer eyewalls.

3.8 Climate Change & Tropical Cyclone Intensity

The relationship between climate change and tropical cyclone behaviour is nuanced. The scientific consensus from Knutson et al. (2020) and the IPCC AR6:

High confidence: the proportion of intense (Cat 4\(\text{--}\)5) TCs has increased and will continue to increase
High confidence: TC-associated precipitation rates increase at ~7\(\text{--}\)14%/°C (Clausius-Clapeyron; see Module 4)
Medium confidence: global total TC frequency may decrease slightly (\(\sim\)10\(\text{--}\)20%), but this masks a shift toward more intense storms
Medium confidence: rapid intensification events become more frequent
High confidence: storm surge risk increases due to sea level rise compounding wind-driven surge
Low confidence: poleward migration of TC tracks (latitude of lifetime maximum intensity shifting ~1° per decade in Western Pacific)

The power dissipation index (PDI), which integrates \(V_{\max}^3\) over the storm lifetime, has been shown to track SST anomalies closely in the Atlantic. Emanuel (2005) demonstrated that PDI roughly doubled over the preceding 30 years, consistent with the PI theory prediction. The ecological consequences of intensifying TCs, from coral reef destruction to forest disturbance regimes, are explored in ourClimate & Biodiversity course.

An important recent finding is the increase in TC translation speed slowdown. Slower-moving storms produce more rainfall at any given location, increasing flood risk. Hurricane Harvey (2017) stalled over Houston, producing \(\sim\)1500 mm of rainfall over 4 days. The stalling was linked to a weakened steering flow, which some studies attribute to Arctic amplification reducing the equator-to-pole temperature gradient and weakening midlatitude westerlies.

References

Emanuel, K. A. (1986). An air-sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. Journal of the Atmospheric Sciences, 43(6), 585-605.

Emanuel, K. A. (1988). The maximum intensity of hurricanes. Journal of the Atmospheric Sciences, 45(7), 1143-1155.

Holland, G. J. (1980). An analytic model of the wind and pressure profiles in hurricanes. Monthly Weather Review, 108(8), 1212-1218.

Tang, B. & Emanuel, K. (2012). A ventilation index for tropical cyclones. Bulletin of the American Meteorological Society, 93(12), 1901-1912.

Knutson, T. R., et al. (2020). Tropical cyclones and climate change assessment: Part II. Bulletin of the American Meteorological Society, 101(3), E303-E322.

Irish, J. L., Resio, D. T., & Ratcliff, J. J. (2008). The influence of storm size on hurricane surge. Journal of Physical Oceanography, 38(9), 2003-2013.

Kossin, J. P., et al. (2020). Global increase in major tropical cyclone exceedance probability over the past four decades. Proceedings of the National Academy of Sciences, 117(22), 11975-11980.

Wing, A. A., Emanuel, K., Holloway, C. E., & Muller, C. (2017). Convective self-aggregation in numerical simulations: a review. Surveys in Geophysics, 38, 1173-1197.

Emanuel, K. A. (2005). Increasing destructiveness of tropical cyclones over the past 30 years. Nature, 436, 686-688.

Kossin, J. P. (2018). A global slowdown of tropical-cyclone translation speed. Nature, 558, 104-107.

Willoughby, H. E., Clos, J. A., & Shoreibah, M. G. (1982). Concentric eye walls, secondary wind maxima, and the evolution of the hurricane vortex. Journal of the Atmospheric Sciences, 39(2), 395-411.